The Mexicans are usually overwhelming favourites against smaller football nations like the All Whites at their imposing, 105,000-capacity Azteca Stadium in Mexico City.
But El Tri won just once in five home games in the final qualifying round of the North, Central America and Caribbean (CONCACAF) group, finishing fourth out of six teams.
While USA, Costa Rica and Honduras won direct tickets to Brazil, Mexico barely secured the play-off - a surprisingly poor showing for a powerhouse that was last absent from a World Cup in 1990.
The dismal results led the football federation to sack three managers in six weeks, finally handing the reins to Miguel Herrera last month for the two-game play-off.
The outspoken coach, who led Club America to a domestic title this year, voiced confidence that his squad can win at home and in the return leg in New Zealand on 20 November.
"The first thing we said was that we shouldn't think about what happened in the past because that's something we can't resolve," Herrera said.
Herrera has shaken things up since taking over, calling up only players from the domestic league while leaving out European-based stars, including Manchester United's Javier "Chicharito" Hernandez and Villarreal's Giovani dos Santos.
He said Mexico would seek to exploit New Zealand's loss of its captain, defender Winston Reid, who was ruled out of the play-off with an ankle injury suffered playing for his English Premier League club West Ham.
Fallon famously scored the goal against Bahrain in the second leg of their play-off that sent New Zealand to the 2010 World Cup in South Africa.
This time, however, the All Whites face a much stronger side that played in 14 of the last 19 World Cups. New Zealand have appeared in only two World Cups: Spain in 1982 and South Africa in 2010.
Herbert said he thought Wednesday's match was "one of the hardest, if not the hardest" New Zealand had ever faced.